ME321 Kinematics and Dynamics of Machines Steve Lambert Mechanical Engineering, U of Waterloo 1/18/2019

Vibrations Vibration: repeated or cyclic motion. usually small scale (small amplitude), usually undesirable, and can lead to noise, fatigue failures, etc. 1/18/2019

Natural Frequencies Preferred frequencies of vibration for a system. Excitation (loading) at these frequencies can lead to resonance. The excitation frequency is the frequency of the applied loading. Examples include shock loading, wave loading, out-of-balance shafts, etc. 1/18/2019

Damping Energy ‘absorption’ (actually transfer to heat energy) Reduces the amplitude of vibration with time and can result from ‘natural damping’ of a material, friction (Coulomb), air resistance, or fluid viscosity, for example. A shock absorber is designed to produce damping. 1/18/2019

Mass Spring Damper System k c x F(t) F(t): excitation force (N) m: mass (kg) k: spring stiffness (N/m) c: viscous damping coefficient (Ns/m) To continue, we draw the free body diagram for the mass 1/18/2019

Free Body Diagram From force equilibrium: or: This is the governing differential equation for forced vibration 1/18/2019

Forced Displacement 1/18/2019

Types of Loading Transient - represented by non-zero initial conditions for displacement and velocity Steady-state - represented by sinusoidal loading, or a combination of sinusoidal loading using Fourier Series analysis Random - represented using Fourier Series, so solution technique similar to steady-state 1/18/2019

Types of Solution All types of loading can be handled by just two solutions Unforced Vibration: F(t) = 0 Forced Vibration: F(t) = F0 sin  t Forced vibration solution requires the unforced (natural) solution to obtain the general solution, and we then add the particular solution 1/18/2019